Geometry and Arithmetic

pp: 113–123

DOI: 10.4171/119-1/7

A remark on a conjecture of Paranjape and Ramanan

Friedrich Eusen[1] and Frank-Olaf Schreyer

(1) Düsseldorf, Germany

In this note, we show that the spaces of global sections of exterior powers of a globally generated line bundle on a curve are not necessarily spanned by locally decomposable sections.
The examples are based on the study of generic syzygy varieties. An application of these varieties is a short proof of Mukai's theorem that every smooth curve of genus 7 and Clifford index 3 arises as the intersection of the spinor variety $S \subset \mathbb P^{15}$ with a transversal $\mathbb P^6$.